Branchless O(1) Shortcut Architecture for Collatz Simulations

Abstract

This study introduces a high-performance, branchless computing framework designed to eliminate conditional logic bottlenecks and intermediate parity checks in Collatz trajectory simulations. By decomposing the 3n+1 operator into modular G1/G2 parity components, we derive a general analytical shortcut operator, f(n,k) = (9n + 3 + 2^k) / 2^k, reducing intermediate step evaluation to O(1) constant time. The architecture integrates an A1-A3 layer-depth hierarchy with the A4 macro-conservation law to model trajectory bounds without iterative execution. Implemented via low-level bit-masking and instruction fusion, the framework prevents CPU pipeline stalls and GPU thread divergence. Empirical validation demonstrates high-throughput evaluation for massive integers up to N = 10^10000 + 1 (~33,219 bits) in 0.8 seconds. Code Availability: https://github.com/AlperPe/collatz-graphics